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We describe a simple scheme for the implementation and control of effectivespin-spin interactions in self-assembled crystals of cold polar molecules. Inour scheme spin states are encoded in two long-lived rotational states of themolecules and coupled via state dependent dipole-dipole forces to the latticevibrations. We show that by choosing an appropriate time dependent modulationof the induced dipole moments the resulting phonon-mediated interactionscompete with the direct dipole-dipole coupling and lead to long-range andtunable spin-spin interaction patterns. We illustrate how this technique can beused for the generation of multi-particle entangled spin states and theimplementation of spin models with longe-range and frustrated interactionswhich exhibit non-trivial phases of magnetic ordering.

Long-range and frustrated spin-spin interactions in crystals of cold polar molecules